Critical fluctuations in the quantum parametric oscillator

We calculate the scale of the critical fluctuations in the quantum parametric oscillator at threshold. This allows us to calculate asymptotically the size of both the depletion of the pump mode and the amount of squeezing produced in the signal mode for large system size. In this driven, open quantum system, there is no known solution to the equilibrium density matrix. For this reason these results cannot be obtained using standard canonical ensemble methods. The calculations are compared to time dependent quantum stochastic simulations of the fully nonlinear coupled mode equations. We find that the critical squeezing variance in the output spectrum will scale as ${\mathit{N}}^{\mathrm{-}1/2}$, giving the feature that these fluctuations are reduced rather than enhanced near the critical point.